Theorem appendinj2 ≪ | index | src | ≫ 

theorem appendinj2 (l1 l2 r1 r2: nat):
  $ len l2 = len r2 -> (l1 ++ l2 = r1 ++ r2 <-> l1 = r1 /\ l2 = r2) $;
StepHypRefExpression
1 appendinj1 
len l1 = len r1 -> (l1 ++ l2 = r1 ++ r2 <-> l1 = r1 /\ l2 = r2)
2 addcan1 
len l1 + len l2 = len r1 + len l2 <-> len l1 = len r1
3 appendlen 
len (l1 ++ l2) = len l1 + len l2
4 appendlen 
len (r1 ++ r2) = len r1 + len r2
5 anr 
len l2 = len r2 /\ l1 ++ l2 = r1 ++ r2 -> l1 ++ l2 = r1 ++ r2
6 5 leneqd 
len l2 = len r2 /\ l1 ++ l2 = r1 ++ r2 -> len (l1 ++ l2) = len (r1 ++ r2)
7 3, 4, 6 eqtr3g 
len l2 = len r2 /\ l1 ++ l2 = r1 ++ r2 -> len l1 + len l2 = len r1 + len r2
8 anl 
len l2 = len r2 /\ l1 ++ l2 = r1 ++ r2 -> len l2 = len r2
9 8 eqcomd 
len l2 = len r2 /\ l1 ++ l2 = r1 ++ r2 -> len r2 = len l2
10 9 addeq2d 
len l2 = len r2 /\ l1 ++ l2 = r1 ++ r2 -> len r1 + len r2 = len r1 + len l2
11 7, 10 eqtrd 
len l2 = len r2 /\ l1 ++ l2 = r1 ++ r2 -> len l1 + len l2 = len r1 + len l2
12 2, 11 sylib 
len l2 = len r2 /\ l1 ++ l2 = r1 ++ r2 -> len l1 = len r1
13 1, 12 syl 
len l2 = len r2 /\ l1 ++ l2 = r1 ++ r2 -> (l1 ++ l2 = r1 ++ r2 <-> l1 = r1 /\ l2 = r2)
14 13, 5 mpbid 
len l2 = len r2 /\ l1 ++ l2 = r1 ++ r2 -> l1 = r1 /\ l2 = r2
15 anrl 
len l2 = len r2 /\ (l1 = r1 /\ l2 = r2) -> l1 = r1
16 anrr 
len l2 = len r2 /\ (l1 = r1 /\ l2 = r2) -> l2 = r2
17 15, 16 appendeqd 
len l2 = len r2 /\ (l1 = r1 /\ l2 = r2) -> l1 ++ l2 = r1 ++ r2
18 14, 17 ibida 
len l2 = len r2 -> (l1 ++ l2 = r1 ++ r2 <-> l1 = r1 /\ l2 = r2)

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)